8.3 Profit & Loss, Interest, Discounting, Speed-Distance & Averages

8.3 Profit & Loss, Interest, Discounting, Speed-Distance & Averages

Core Principle: Quantitative aptitude problems in commercial arithmetic and kinetics are governed by invariant conservation laws: total value conserved across transactions, total work conserved across contributors, and total distance conserved across journeys. Identifying the invariant baseline—whether Cost Price, Total Work, or Total Displacement—transforms multi-step word problems into clean, solvable algebraic equations.

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1. Profit, Loss, and Commercial Discounting

Commercial arithmetic problems in CG SET Paper I revolve around the dynamic relationship between three monetary parameters:

• Cost Price (CP): The net expenditure incurred by a merchant to purchase or manufacture a product.
• Selling Price (SP): The actual transaction price at which the good is exchanged with the customer.
• Marked Price (MP / List Price): The catalog price affixed to the item before any promotional concessions.

1.1 Invariant Rules of Profit, Loss, and Discount

$\text{Profit} = SP - CP \quad (\text{when } SP > CP)$ $\text{Loss} = CP - SP \quad (\text{when } CP > SP)$

$\text{Profit } \% = \left(\frac{SP - CP}{CP}\right) \times 100\% = \left(\frac{\text{Profit}}{CP}\right) \times 100\%$ $\text{Loss } \% = \left(\frac{CP - SP}{CP}\right) \times 100\% = \left(\frac{\text{Loss}}{CP}\right) \times 100\%$

Non-Negotiable Examination Rule: Profit percentage and Loss percentage are ALWAYS calculated upon the Cost Price (CP) unless an exam question explicitly instructs otherwise.

Direct Scale Multipliers for Rapid Calculation

Instead of multi-step unitary calculations, utilize scale multipliers:

• $SP = CP \times \left(1 + \frac{P\%}{100}\right) = CP \times \left(\frac{100 + P\%}{100}\right)$
• $SP = CP \times \left(1 - \frac{L\%}{100}\right) = CP \times \left(\frac{100 - L\%}{100}\right)$
• $CP = \frac{SP \times 100}{100 + P\%} \quad \text{or} \quad CP = \frac{SP \times 100}{100 - L\%}$

1.2 Discounting Mechanics and the CP-MP Bridge

Discount is a reduction conceded to the buyer. Discount is ALWAYS calculated upon the Marked Price (MP):

$\text{Discount} = MP - SP$ $\text{Discount } \% = \left(\frac{MP - SP}{MP}\right) \times 100\% \implies SP = MP \times \left(\frac{100 - D\%}{100}\right)$

The Unified CP-MP Master Equation

Because Selling Price ($SP$) bridges both transactions:

$SP = CP \times \left(\frac{100 + P\%}{100}\right) = MP \times \left(\frac{100 - D\%}{100}\right)$

$\frac{MP}{CP} = \frac{100 + P\%}{100 - D\%}$

(If the merchant incurs a loss $L\%$, replace $+ P\%$ with $- L\%$).

Successive Discounts

When a seller offers consecutive trade discounts of $d_1\%$ and $d_2\%$:

$\text{Single Equivalent Discount} = \left(d_1 + d_2 - \frac{d_1 \cdot d_2}{100}\right)\%$

• Demonstration: Two successive discounts of $20\%$ and $10\%$: $\text{Equivalent Discount} = 20 + 10 - \frac{200}{100} = 30 - 2 = 28\% \quad (\text{NOT } 30\%!).$

1.3 Dishonest Dealer & False Weights

A merchant claims to sell commodities at cost price ($SP = CP$), but uses a false weight that registers $W$ grams while delivering only $w$ grams ($w < W$):

$\text{Gain } \% = \left(\frac{\text{Error}}{\text{False Weight}}\right) \times 100\% = \left(\frac{\text{True Weight} - \text{False Weight}}{\text{False Weight}}\right) \times 100\%$

• Demonstration: A dealer sells sugar at cost price but uses a 900-gram weight for 1 kilogram (1000 grams): $\text{Gain } \% = \frac{1000 - 900}{900} \times 100\% = \frac{100}{900} \times 100\% = \frac{1}{9} \times 100\% = 11.11\%$

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2. Simple Interest (SI) vs. Compound Interest (CI)

Interest is the rental fee paid for the use of borrowed capital ($P$) over time ($T$) at an agreed annual percentage rate ($R\%$).

2.1 Simple Interest (SI): The Linear Growth Model

Simple interest accrues strictly on the original principal throughout the entire duration. Interest generated in prior periods is never capitalized:

$SI = \frac{P \times R \times T}{100}$ $\text{Maturity Amount } (A) = P + SI = P \left(1 + \frac{R \cdot T}{100}\right)$

The Linear Doubling and Tripling Rules

• A sum doubles ($A = 2P \implies SI = P$) in: $T = \frac{100}{R}$ years.
• If a sum becomes $n$ times its original principal in $T$ years at simple interest: $R = \frac{(n - 1) \times 100}{T} \quad \text{or} \quad T = \frac{(n - 1) \times 100}{R}$

2.2 Compound Interest (CI): The Exponential Growth Model

Under compound interest, accrued interest is added to the principal at the end of each conversion period. Subsequent interest is calculated on this expanding base:

$A = P \left(1 + \frac{R}{100}\right)^T$ $CI = A - P = P \left[\left(1 + \frac{R}{100}\right)^T - 1\right]$

Compounding Frequency Conversions

• Compounded Semi-Annually (Half-Yearly): Rate becomes $\frac{R}{2}\%$, conversion periods become $2T$: $A = P \left(1 + \frac{R/2}{100}\right)^{2T}$
• Compounded Quarterly: Rate becomes $\frac{R}{4}\%$, conversion periods become $4T$: $A = P \left(1 + \frac{R/4}{100}\right)^{4T}$

The Rule of 72

An investment compounded annually doubles in approximately $\frac{72}{R}$ years (e.g., at $8\%$ compound interest, doubling requires $\frac{72}{8} \approx 9$ years).

2.3 Master Difference Formulas: CI vs. SI

Examination problems frequently specify the numerical difference ($D$) between compound interest and simple interest on the same principal for identical periods:

1. For 1 Year (Compounded Annually): $D_1 = CI_1 - SI_1 = 0$
2. For 2 Years: $D_2 = CI_2 - SI_2 = P \left(\frac{R}{100}\right)^2$
3. For 3 Years: $D_3 = CI_3 - SI_3 = P \left(\frac{R}{100}\right)^2 \left(\frac{300 + R}{100}\right)$

2.4 Present Worth, True Discount and Banker's Discount

The syllabus word "discounting" covers two ideas: the trade discount on a marked price (section 1.2) and financial discounting, which runs interest backwards to find what a future sum is worth today. Suppose a sum $A$ is due $T$ years from now and money earns simple interest at $R\%$ a year.

• Present Worth (PW): the sum that, invested today at $R\%$ for $T$ years, grows to $A$: $PW = \frac{100 \times A}{100 + RT}$
• True Discount (TD): the difference between the amount due and its present worth. It equals the simple interest on the present worth: $TD = A - PW = \frac{A \times R \times T}{100 + RT} = \frac{PW \times R \times T}{100}$
• Banker's Discount (BD): a bank that buys (discounts) a bill before its due date deducts simple interest on the face value $A$, not on the present worth: $BD = \frac{A \times R \times T}{100}$
• Banker's Gain (BG): the bank's extra gain over the true discount, which equals the simple interest on the true discount: $BG = BD - TD = \frac{TD^2}{PW}$
• Compound discounting: with annual compounding at rate $r$ (as a decimal), the present value of $A$ due after $n$ years is $PV = \frac{A}{(1 + r)^n}$

Worked example: A bill of ₹1,100 is due in 1 year and money earns 10% simple interest a year.

$PW = \frac{100 \times 1{,}100}{100 + 10 \times 1} = \frac{110{,}000}{110} = 1{,}000$

So the true discount is ₹1,100 − ₹1,000 = ₹100. The banker's discount is $\frac{1{,}100 \times 10 \times 1}{100} = 110$, that is ₹110. The banker's gain is ₹110 − ₹100 = ₹10, which matches $\frac{TD^2}{PW} = \frac{100^2}{1{,}000} = 10$.

Exam cue: banker's discount is always larger than true discount for the same bill, because it charges interest on the larger face value.

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3. Time, Speed, and Distance (TSD) Mechanics

Motion problems model the relationship between spatial distance ($D$), velocity ($S$), and elapsed time ($T$):

$\text{Distance} = \text{Speed} \times \text{Time} \implies S = \frac{D}{T}, \quad T = \frac{D}{S}$

3.1 Unit Conversion Invariants

Mixing kilometer-per-hour and meter-per-second units is the most frequent calculation blunder in CG SET Paper I:

$1\text{ km/h} = \frac{1{,}000\text{ meters}}{3{,}600\text{ seconds}} = \frac{5}{18}\text{ m/s}$ $1\text{ m/s} = \frac{18}{5}\text{ km/h}$

• To convert $\text{km/h}$ to $\text{m/s}$: Multiply by $\frac{5}{18}$ (e.g., $72\text{ km/h} \times \frac{5}{18} = 20\text{ m/s}$).
• To convert $\text{m/s}$ to $\text{km/h}$: Multiply by $\frac{18}{5}$ (e.g., $25\text{ m/s} \times \frac{18}{5} = 90\text{ km/h}$).

3.2 Average Speed Formulations

Average speed is never the simple arithmetic mean of speeds across different legs:

$\text{Average Speed} = \frac{\text{Total Distance Traveled}}{\text{Total Time Taken}}$

• Case 1: Equal Distances (The Harmonic Mean): If a traveler covers distance $D$ at speed $x$ and returns the same distance $D$ at speed $y$: $\text{Average Speed} = \frac{2xy}{x + y}$
• Case 2: Equal Time Intervals (The Arithmetic Mean): If a vehicle travels for duration $t$ at speed $x$ and subsequent duration $t$ at speed $y$: $\text{Average Speed} = \frac{x + y}{2}$

3.3 Relative Speed Dynamics

When two physical bodies move simultaneously:

• Moving in Opposite Directions (Towards or Away): Their velocities combine to close or open the distance faster: $S_{\text{relative}} = S_1 + S_2$
• Moving in the Same Direction: The faster body gains upon the slower body at their net velocity difference: $S_{\text{relative}} = |S_1 - S_2|$

3.4 Train Crossing Mechanics

1. Train Crossing a Point Object of Negligible Length (Pole, Tree, Person): $\text{Distance Covered} = L_{\text{train}}, \quad \text{Time} = \frac{L_{\text{train}}}{S_{\text{train}}}$
2. Train Crossing an Object of Extended Physical Length (Platform, Bridge, Tunnel): $\text{Distance Covered} = L_{\text{train}} + L_{\text{platform}}, \quad \text{Time} = \frac{L_{\text{train}} + L_{\text{platform}}}{S_{\text{train}}}$
3. Two Trains Crossing Each Other: $\text{Total Distance} = L_1 + L_2$ $\text{Time (Opposite Directions)} = \frac{L_1 + L_2}{S_1 + S_2}, \quad \text{Time (Same Direction)} = \frac{L_1 + L_2}{|S_1 - S_2|}$

3.5 Boats and Streams (River Kinetics)

Let speed of the boat in still water be $u$ and speed of the water current/stream be $v$:

• Downstream Speed (with the stream): $D = u + v$
• Upstream Speed (against the stream): $U = u - v$
• Derived speeds: $u = \frac{D + U}{2}, \quad v = \frac{D - U}{2}$

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4. Time and Work & Pipes and Cisterns

4.1 The Reciprocal Work-Rate Framework

Time and work problems are grounded in reciprocal rates. If an individual can complete a task in $d$ days, their 1-day work rate is $\frac{1}{d}$ of the total task:

$\text{Work Done} = \text{Rate} \times \text{Time} \implies \text{Total Work} = 1$

• Combined Work of A and B: If person A takes $A$ days and person B takes $B$ days: $\text{Combined 1-day Rate} = \frac{1}{A} + \frac{1}{B} = \frac{A + B}{A \cdot B}$ $\text{Total Time Taken Together} = \frac{A \cdot B}{A + B}\text{ days}$
• Three Contributors (A, B, C): $\text{Total Time} = \frac{A \cdot B \cdot C}{AB + BC + CA}\text{ days}$

The Work Equivalence Equation

When varying teams of workers ($M$) work for days ($D$) and daily hours ($H$) to produce work output ($W$):

$\frac{M_1 \cdot D_1 \cdot H_1}{W_1} = \frac{M_2 \cdot D_2 \cdot H_2}{W_2}$

4.2 Pipes and Cisterns

Pipes and cisterns problems operate identically to time and work, with the addition of negative rates:

• Inlet Pipe (Fills the tank): Positive work rate $= +\frac{1}{A}$ per hour.
• Outlet Pipe / Leak (Empties the tank): Negative work rate $= -\frac{1}{B}$ per hour.
• Net Filling Rate: $\text{Net Rate} = \frac{1}{A} - \frac{1}{B} = \frac{B - A}{AB} \implies \text{Time to Fill Tank} = \frac{AB}{B - A}$

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5. Averages and Arithmetic Means

The average (arithmetic mean) represents the balanced central measure of a distribution of numerical observations:

$\text{Average } (\bar{x}) = \frac{\text{Sum of All Observations}}{\text{Number of Observations}} = \frac{\sum x_i}{N}$ $\text{Total Sum} = N \times \bar{x}$

5.1 Weighted Average

When groups with varying sizes ($n_1, n_2, \dots$) have distinct averages ($A_1, A_2, \dots$):

$\bar{x}_w = \frac{n_1 A_1 + n_2 A_2 + \dots + n_k A_k}{n_1 + n_2 + \dots + n_k}$

5.2 Arithmetic Properties of Sequences

• Average of the first $n$ natural numbers ($1, 2, \dots, n$): $\frac{n + 1}{2}$
• Average of the first $n$ even natural numbers ($2, 4, \dots, 2n$): $n + 1$
• Average of the first $n$ odd natural numbers ($1, 3, \dots, 2n - 1$): $n$
• Average of any Arithmetic Progression: $\frac{\text{First Term} + \text{Last Term}}{2}$

5.3 The Replacement Theorem

When a group of $N$ items undergoes a replacement where an old item of value $W_{\text{old}}$ is substituted by a new item $W_{\text{new}}$, altering the group's average by $\Delta\bar{x}$:

$W_{\text{new}} = W_{\text{old}} \pm (N \times \Delta\bar{x})$

• Demonstration: A research committee of 10 scholars has an average age. When a 50-year-old professor retires and is replaced by a younger researcher, the committee's average age decreases by 2 years ($N = 10, \Delta\bar{x} = -2$): $W_{\text{new}} = 50 - (10 \times 2) = 50 - 20 = 30\text{ years}.$

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6. Comprehensive Formula Reference Table for Chapter 8

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7. Exam Traps & Common Calculation Blunders

• Trap 1: Calculating Profit/Loss upon Selling Price: A question states "a vendor earns 20% profit on selling price". If $SP = 100$, then Profit $= 20$, which implies $CP = 80$. The actual true profit percentage is $\frac{20}{80} \times 100\% = 25\%$. Never confuse profit on SP with profit on CP.
• Trap 2: Arithmetic Averaging of Speeds across Equal Distances: If a car travels to a city at 60 km/h and returns at 40 km/h, the average speed is not 50 km/h. It is the harmonic mean: $\frac{2 \times 60 \times 40}{60 + 40} = \frac{4800}{100} = 48\text{ km/h}$.
• Trap 3: Omitting Platform Length in Train Problems: When a train crosses a bridge or platform, the total distance traveled is $L_{\text{train}} + L_{\text{platform}}$. Forgetting to add the train's own length produces an erroneous, truncated answer.
• Trap 4: Forgetting Compounding Conversion Periods: In semi-annual compound interest problems, candidates frequently halve the interest rate ($R/2$) but forget to double the time periods ($2T$), leading to gross compounding errors.